Sapphire, which is a single crystal form of alumina, has crystal lattice constants similar to those of gallium nitride (GaN), which is a common material for light emitting diodes (LEDs). Sapphire is thus widely used as a substrate for growing GaN crystals for LEDs. Sapphire substrates for growing GaN crystals include, in addition to substrates with a precisely flat surface (main surface), sapphire substrates with an irregular main surface or with an appropriately roughened main surface for efficient growth of buffer layers to reduce defects caused by dislocations of GaN or other crystals. For example, Japanese Unexamined Patent Application Publication No. 11-274568 (Patent Literature 1) describes sapphire substrates having a main surface with recesses formed by reactive ion etching or having a main surface roughened by mechanical polishing using relatively coarse abrasive grains.